Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: $ x^{2} + 78 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 27 + 73\cdot 79 + 47\cdot 79^{2} + 28\cdot 79^{3} + 7\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 44\cdot 79 + 2\cdot 79^{2} + 78\cdot 79^{3} + 4\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 a + 3 + \left(67 a + 48\right)\cdot 79 + \left(55 a + 36\right)\cdot 79^{2} + \left(23 a + 40\right)\cdot 79^{3} + \left(73 a + 78\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 a + 30 + \left(11 a + 9\right)\cdot 79 + \left(23 a + 25\right)\cdot 79^{2} + \left(55 a + 8\right)\cdot 79^{3} + \left(5 a + 49\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 a + 26 + \left(56 a + 62\right)\cdot 79 + \left(20 a + 40\right)\cdot 79^{2} + \left(47 a + 27\right)\cdot 79^{3} + \left(27 a + 58\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 a + 66 + \left(22 a + 78\right)\cdot 79 + \left(58 a + 4\right)\cdot 79^{2} + \left(31 a + 54\right)\cdot 79^{3} + \left(51 a + 38\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3,4,6,5)$ |
| $(1,3)(2,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,3)(2,4)(5,6)$ | $-2$ |
| $15$ | $2$ | $(1,5)(4,6)$ | $0$ |
| $20$ | $3$ | $(1,3,6)(2,4,5)$ | $1$ |
| $30$ | $4$ | $(1,6,5,4)$ | $0$ |
| $24$ | $5$ | $(1,4,3,2,6)$ | $-1$ |
| $20$ | $6$ | $(1,2,3,4,6,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
